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What's going on with this 5-line proof of Fermat's Last Theorem? [duplicate]

I'm reading a book on the Philosophy of Mathematics, and the author gave a "5-line proof" of Fermat's Last Theorem as a way to introduce the topic of inconsistency in set theory and logic. The author ...
0
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0answers
58 views

How mathematics would be different if the first derivations, conjectures and theorems would be others? [on hold]

I've realised that mathematics is nothing else that an implication of some assumptions (plus the assumptions themselves, of course). We have axioms and we derive new "things", new rules, ideas, ...
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51 views

How to become fluent at reading math formulas? [on hold]

As part of my studies, oftentimes I need to read research publications which contain mathematical formulas. Whenever I have to do that, I feel discouraged. Somehow I can not comprehend the ...
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1answer
47 views

The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
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2answers
47 views

Necessary truth of mathematical proposition.

Take from Possible world- an introduction to logic and its philosophy. p-21 Following quote provide us with necessary definition of what "logically necessary" or as far as i think "necessary truth" ...
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1answer
67 views

Can we take definability and existence as primitive notions of a theory?

One of my friend tries to develop an alternative viewpoint of Set Theory. For this he has taken the terms binary relation, set, existence and definability as primitive notions of his Set Theory. After ...
5
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6answers
349 views

Do different notations imply different properties of a number?

I had an argument with a friend of mine and I'd be glad if someone could clarify things a little bit. So, let's say we have an integer, eight or seventeen, for example, doesn't matter. It has all the ...
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1answer
70 views

A philosophical question on probability theory [closed]

This question is philosophical in nature. The example is taken from theology, but one may invent more examples, including these more scientific than mine. Nevertheless it is a valid mathematical issue....
2
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1answer
65 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
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2answers
122 views

When are quantities outside of the real numbers considered equal, and when do they exist?

I know of the complex number $i$ and it's existence as the result of invalid square rooting (the square root of negative one does not exist inside the real numbers), but other than complex numbers, ...
51
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10answers
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Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
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0answers
25 views

Tensors as mathematical objects

Continuing my journey to understand Tensors, Maxwell's equations. Here is my current understanding. Is it correct? Tensors are mathematical objects, i.e., an entity in mathematical reality or a ...
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4answers
677 views

What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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1answer
90 views

Is it circular to define the Von Neumann universe using “sets”?

I was just reading the Wikipedia page on the Von Neumann universe, where it is stated that this universe "is often used to provide an interpretation or motivation of the axioms of ZFC." However, later ...
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6answers
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Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
2
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1answer
60 views

Cauchy's real line and math philosophy till XIX

I have to write an essay concerning philosophy of mathematics until the end of $XIX$ century. I've heard that the reason why the Cauchy's theorem (if continuous functions $f_n \rightarrow f$ then $f$ ...
2
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0answers
69 views

Is Leibnizian calculus embeddable in first order logic?

We just published an article making what we feel is a plausible case in favor of an affirmative answer in Foundations of Science, see preprint here. The basic argument is that while such a requirement ...
2
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1answer
48 views

Have humans ever used the Log Scale convention in the past rather than the Linear one?

There are many examples where our senses are based off of log scales such as volume of a noise, ability to guess (i.e.) plus or minus a power of 10 with Fermi, and even when we measure pain on 1 to 10 ...
3
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3answers
144 views

On “why” questions in mathematics

In response to the question How would one be able to prove mathematically that $1+1 = 2$?, Asaf Karagila explains: In a more general setting, one needs to remember that $0,1,2,3,…$ are just ...
5
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2answers
81 views

On logic vs information theory

If the statements All crows are black and All non black things are non crows are equal, then why is the former so much easier to communicate by giving examples? What implications does this ...
27
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9answers
2k views

Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
3
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0answers
70 views

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm ...
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1answer
45 views

Different Mathematics

Hey I am a high school student who is very interested in the philosophy of mathematics. I was watching this talk by Stephen Wolfram about whether or not mathematics is invented or discovered. In it he ...
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1answer
22 views

A question about the real line and the Dirichlet function.

Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points). Here's my reasoning: ...
5
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4answers
103 views

Should we or should we not take $1$ as a prime number? [duplicate]

I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime ...
2
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3answers
38 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...
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mathematical terms with fractions and variables - usage in daily life?

what usage do algebraic fractions (monomial or polynomial) have in our life? Are there specific professions that deal with them now and then? Where exactly in technology do they have their very own ...
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$Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ for any intuitionistic theory T

It's easy to notice that for any intuitionistic theory T: $Con(T) + T\vdash \neg\neg A$ implies $Con(T+A)$ $Con(T) + T\vdash \neg\forall x\neg A$ implies $Con(T+\exists x A)$ where $Con(T)$ means, ...
3
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2answers
62 views

Scalar multiplication as a special form of matrix multiplication

Question What do we gain or lose, conceptually, if we consider scalar multiplication as a special form of matrix multiplication? Background The question bothers me since I have been reading about ...
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2answers
54 views

Impossibility of proving a foundation to be consistent

An argument came to my mind that seems really mind-blowing and I haven't found it anywhere. Here's how it goes: We call a formal system F embodied in classical logic a foundation of mathematics when ...
1
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1answer
104 views

Quasi mathematical objects [closed]

I was looking on this post http://www.songho.ca/math/euler/euler.html and I came to the comment that says "i is not a number at all. It is an ill-formed concept. There is a vast difference between a ...
2
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1answer
37 views

Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity'...
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2answers
53 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
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1answer
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Developments from Charles Peirce's logic diagrams?

These last weeks I have been revisiting Charles Sanders Peirce's logical or thought diagrams (what he called, alpha, beta and gamma diagramms) and I found many of them highly interesting. Some ...
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3answers
167 views

What exactly are the numbers we use everyday?

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its ...
2
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2answers
93 views

What is the simplest mathematical object? [closed]

What is the simplest mathematical object? I am talking about mathematics in the most abstract way possible, and not as some concrete axiomatic theory (e.g. foundational ones, like ZFC). After a lot ...
5
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0answers
94 views

How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...
8
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3answers
169 views

How should a mathematically-inclined person learn descriptive statistics?

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page ...
7
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3answers
168 views

Why might Dieudonne have been “begging the question” by appealing to second-order Peano Axioms?

Following a comment by Peter Smith, I've been reading A. R. D. Mathias's paper The Ignorance of Bourbaki. Parts of the paper are above my head, but I understand it well enough for my own amateurish ...
2
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2answers
38 views

Why can't a Hilbert curve be used to put the real numbers into a listable format?

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from. From my ...
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Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
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63 views

No Proof, Just Luck

I just read about the Goldbach Conjecture and it got me thinking about probabilities. Supposing that prime numbers are somewhat randomly distributed) then if we calculate the odds of a given even ...
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41 views

Natural numbers, divisors, primes and their generalized means

Let div, nat and pri the finite sequences given in increasing order for an integer $n\geq 1$ of its divisors $1=d_1<d_2<\ldots d_{\sigma_0(n)}=n$, the first $n$ natural numbers, and the first $n$...
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47 views

How can we recognize if something is a number?

There are formal definitions of various types of numbers; natural numbers, real numbers, ordinal numbers, cardinals etc. And we all regard them as some type of number. Are there properties that are ...
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1answer
42 views

What is this ontological position called?

If one believes that certain 'abstract' mathematics-like concepts do exist, yet the mathematics we construct and develop as humans are only approximations of those real concepts, approximations shaped ...
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5answers
736 views

How does one refute this ultrafinitist argument?

From Wikipedia: Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. In classical mathematics the natural numbers are defined as $0$ ...
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0answers
54 views

Can I just make this function up?

The Lambert W function was made to solve the problem $xe^x=k$ for $x$, which is given as $x=W(k)$. Could I just make a function $x=F(k)$ which solves $x\cos(x)=k$? Even though the solution has an ...
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Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...